Ομάδα SU(2)
Ομάδα SU(2) Group thumb|300px| [[Ομαδοθεωρία ---- Αλγεβρική Ομάδα Γενική Γραμμική Ομάδα Ορθογώνια Ομάδα Μοναδιακή Ομάδα ---- Μαθηματική Αναπαράσταση Μαθηματική Μήτρα ]] - Μία Ομάδα. - Ανήκει στην κατηγορία των μοναδιακών ομάδων. Ετυμολογία Η ονομασία "ομάδα" σχετίζεται ετυμολογικά με την λέξη '"ομού". Εισαγωγή The Lie Algebras of SU(2) and SO(3) are isomorphic This means that SU(2) and SO(3) are locally isomorphic. (This does not mean that SU(2) and SO(3) are isomorphic) SU(2) is actually a double cover of SO(3) and there is a 2→1 surjective homeomorphism from SU(2) to SO(3) Spin 1/2 particles, or fermions, need to be rotated 720º in order to come back to the same state. Σημείωση: not every representation of SU(2) is a representation of SO(3). Η ομάδα SU(2) is the double cover of SO(3), and SU(2) is isomorphic to the coset SO(3)/Z2. Elements of SU(2) are 2x2 complex matrices. If to each matrix A ∈ SU(2) you assing the transformation x ↦ A x of C 2 - then you have the fundamental represantation of SU(2) There is a very nice a natural group homomorphism, call it ρ , ρ: SU(2) → SO(3) . It has the property ρ(A) = ρ(−A). Matrices A and − A are mapped to the same element of SO(3) . Thus the name "double cover". Le groupe SU(2) est explicitement : SU(2) is the following group, : \mathrm{SU}(2) = \left \{ \begin{pmatrix} \alpha&-\overline{\beta}\\ \beta & \overline{\alpha} \end{pmatrix}: \ \ \alpha,\beta\in\mathbf{C}, |\alpha|^2 + |\beta|^2 = 1\right \} ~, where the overline denotes complex conjugation. Now, consider the following map, : \begin{align} \varphi \colon \mathbf{C}^2 &\to \operatorname{M}(2,\mathbf{C}) \\ \varphi(\alpha,\beta) &= \begin{pmatrix} \alpha&-\overline{\beta}\\ \beta & \overline{\alpha}\end{pmatrix},\end{align} where M(2, '''C') denotes the set of 2 by 2 complex matrices. By considering C'2 diffeomorphic to '''R'4 and M(2, '''C) diffeomorphic to R'8, we can see that ''φ is an injective real linear map and hence an embedding. Now, considering the restriction of φ to the 3-sphere (since modulus is 1), denoted ''S''3, we can see that this is an embedding of the 3-sphere onto a compact submanifold of M(2, '''C). However, it is also clear that φ(''S''3) = SU(2). Therefore, as a manifold ''S''3 is diffeomorphic to SU(2) and so ''S''3 is a compact, connected Lie group. The Lie algebra of SU(2) is : \mathfrak{su} (2) = \left \{ \begin{pmatrix} ia & -\overline{z}\\ z & -ia \end{pmatrix}: \ a \in \mathbf{R}, z \in \mathbf{C} \right \} ~. It is easily verified that matrices of this form have trace zero and are antihermitian. The Lie algebra is then generated by the following matrices, : u_1 = \begin{pmatrix} 0 & i\\ i & 0 \end{pmatrix} \qquad u_2 = \begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix} \qquad u_3 = \begin{pmatrix} i & 0\\ 0 & -i \end{pmatrix} ~, which are easily seen to have the form of the general element specified above. These satisfy ''u''3''u''2 = −''u''2''u''3 = −''u''1 and ''u''2''u''1 = −''u''1''u''2 = −''u''3. The commutator bracket is therefore specified by : u_3,u_1=2u_2, \qquad u_1,u_2 = 2u_3, \qquad u_2,u_3 = 2u_1~. The above generators are related to the Pauli matrices by ''u''1 = ''i σ''1,''u''2 = −''i σ''2 and ''u''3 = ''i σ''3. This representation is routinely used in quantum mechanics to represent the spin of fundamental particles such as electrons. They also serve as unit vectors for the description of our 3 spatial dimensions in loop quantum gravity. The Lie algebra serves to work out the representations of U(2). ---- : G = \begin{bmatrix} z & x-iy \\ x+iy & -z \\ \end{bmatrix} : G_0 = \begin{bmatrix} 0 & 0 \\ 0 & 0 \\ \end{bmatrix} : G_1 = \begin{bmatrix} x \\ x \\ \end{bmatrix} : G_2 = \begin{bmatrix} -iy \\ iy \\ \end{bmatrix} : G_3 = \begin{bmatrix} z \\ -z \\ \end{bmatrix} Υποσημειώσεις Εσωτερική Αρθρογραφία *Ομάδα *Ομαδοθεωρία *Ορθογώνια Ομάδα *Μοναδιακή Ομάδα *Ετεροτική Ομάδα *Αναπαράσταση *Μήτρα Βιβλιογραφία * * Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *[ ] *[ ] Category: Μαθηματικές Ομάδες